On a power-type coupled system of Monge-Ampère equations

Abstract

We study an elliptic system coupled by Monge--Amp\`{e}re equations: $$ \begin{cases} \det D^{2}u_{1}={(-u_{2})}^\alpha & \hbox{in $\Omega,$} \det D^{2}u_{2}={(-u_{1})}^\beta & \hbox{in $\Omega,$} u_{1} 0$, $\beta >0$. When $\Omega$ is the unit ball in $\mathbb{R}^{N}$, we use index theory of fixed points for completely continuous operators to get existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on $\alpha$, $\beta$. When $\alpha>0$, $\beta>0$ and $\alpha\beta=N^2$ we also study a~corresponding eigenvalue problem in more general domains.